Abstract

We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system has no solution satisfying condition , where and and and are real-valued functions defined on .

1. Introduction

In 1907, Lyapunov [1] established the first so-called Lyapunov inequality:
if Hill’s equation
has a real solution such that
and the constant 4 in (1) cannot be replaced by a larger number, where is a piecewise continuous and nonnegative function defined on . Since this result has found applications in the study of various properties of solutions such as oscillation theory, disconjugacy, and eigenvalue problems of (2), a large number of Lyapunov-type inequalities were established in the literature which generalized or improved (1); see [1–20].

In 1983, Cheng [3] first obtained the discrete analogy of Lyapunov inequality (1) for the second-order difference equation:
where, and in the sequel, denotes the forward difference operator defined by .

When and , that is, system (4) has a solution satisfying , which is called homoclinic solution, whether one can obtain Lyapunov-type inequalities for (4)? To the best of our knowledge, there are no results.

In 2003, Sh. Guseinov and Kaymakçalan [7] partly generalized the Cheng’s result to the discrete linear Hamiltonian system:
where , , and are real-valued functions defined on and and are not necessarily usual zeros, but rather, generalized zeros. Later, some better Lyapunov-type inequalities for system (5) were obtained in [19, 20].

Very recently, He and Zhang [10] further generalized the result in [19] to the following first-order nonlinear difference system:
where and and , , and are real-valued functions defined on .

When , system (6) reduces to (5). In addition, the special forms of system (6) contain many well-known difference equations which have been studied extensively and have much applications in the literature [21–23], such as the second-order linear difference equation:
and the second-order half-linear difference equation:
where , and are real-valued functions defined on and . Let
then (8) can be written as the form of (6):
where , and , and .

In this paper, we will establish several Lyapunov-type inequalities for systems (5) and (6) if they have a solution satisfying conditions
respectively. Taking advantage of these Lyapunov-type inequalities, we are able to establish some criteria for nonexistence of homoclinic solutions of systems (5) and (6). As we know, there are no results on non-existence of homoclinic solutions for Hamiltonian systems in previous literature.

In this section, we shall establish some Lyapunov-type inequalities for system (6). For the sake of convenience, we list some assumptions on and as follows:(A0), for all , ; (A1), for all , ; (B0), for all ; (B1).

Denote

Theorem 1. Suppose that hypotheses (A0), (B0), and (B1) are satisfied. If system (6) has a solution satisfying
then one has the following inequality:
where .

Proof. Hypothesis (B1) implies that functions and are well defined on . Without loss of generality, we can assume that
From (14) and (B0), one has
It follows from (13), (18), and the Hölder inequality that
From (A0), (17), (19), (20), and the first equation of system (6), we have
Combining (19) with (21), one has
Similarly, it follows from (20) and (22) that
Combining (23) with (24), one has
Now, it follows from (16), (18), and (25) that
By (6), we obtain
Summing the above from to and using (17) and (18), we obtain
which, together with (26), implies that
We claim that
If (30) is not true, then
From (28) and (31), we have
It follows that
Combining (21) with (33), we obtain that
which, together with the second equation of system (6), implies that
Combining the above with (17), one has
Both (34) and (36) contradict with (14). Therefore, (30) holds. Hence, it follows from (29) and (30) that (15) holds.

Corollary 2. Suppose that hypotheses (A1), (B0), and (B1) are satisfied. If system (6) has a solution satisfying (14), then one has the following inequality:
where and in the sequel,

Proof. Obviously, (A1) implies that
and so (A0) holds, and which, together with (B1), implies that . Since
it follows that
which implies that (37) holds.